Pinched-up periodic KPZ fixed point
Jinho Baik, Zhipeng Liu
TL;DR
This work extends conditional limit results for the KPZ fixed point to its periodic counterpart, focusing on a pinch-up event on a ring. By exploiting an explicit multi-time, multi-position Fredholm-determinant formula and a careful asymptotic analysis, the authors identify three regimes determined by the scaling of the ring period $p$ with the pinch-up height $\ell$: (i) large period recovers the line KPZ conditional limit via two independent Brownian bridges, (ii) critical period $p=\mathsf{r}\ell^{-1/4}$ yields a Brownian bridge on the circle $I_{\mathsf{r}}$ with a distance term, and (iii) small period leads to a Brownian-motion–type limit with no spatial dependence. They also derive right-tail asymptotics for the one-point density in these regimes, linking to the GUE Tracy–Widom tail where appropriate. The results illuminate how domain size and relaxation scale influence conditional KPZ fluctuations and provide probabilistic interpretations for the limiting objects in the periodic setting. Overall, the paper advances the universality picture for KPZ in periodic domains and clarifies the geometry of conditioned geodesics via Brownian-bridge and distance-based descriptions on rings.
Abstract
The periodic KPZ fixed point is the conjectural universal limit of the KPZ universality class models on a ring when both the period and time critically tend to infinity. For the case of the periodic narrow wedge initial condition, we consider the conditional distribution when the periodic KPZ fixed point is unusually large at a particular position and time. We prove a conditional limit theorem up to the ``pinch-up" time. When the period is large enough, the result is the same as that for the KPZ fixed point on the line obtained by Liu and Wang in 2022. We identify the regimes in which the result changes and find probabilistic descriptions of the limits.